A direct algorithm to compute the topological Euler characteristic and Chern–Schwartz–MacPherson class of projective complete intersection varieties

Abstract

Let V be a possibly singular scheme-theoretic complete intersection subscheme of P n over an algebraically closed field of characteristic zero. Using a recent result of Fullwood ("On Milnor classes via invariants of singular subschemes", Journal of Singularities) we develop an algorithm to compute the Chern-Schwartz-MacPherson class and Euler characteristic of V . This algorithm complements existing algorithms by providing performance improvements in the computation of the Chern-Schwartz-MacPherson class and Euler characteristic for certain types of complete intersection subschemes of P n .

Computing characteristic classes of subschemes of smooth toric varieties

“…In Theorem 3.7 we give an expression for the c SM class of certain types of complete intersection subschemes of toric varieties of the form X Σ ; this result generalizes Theorem 3.3 of the author [22] and leads to a more efficient algorithm that avoids performing inclusion/exclusion when computing the c SM class in some cases.…”

Probabilistic Saturations and Alt’s Problem

“…. If Z is a projective variety, then the polynomials making up F are homogeneous, C r is replaced by P r−1 , a set of polynomials defining X = V(F ) can be taken to have the same degree (without changing X ⊂ Z), and the numbers g i (X, Y ) are the projective degrees of X in Y (see [19] for the general case as well as [20,Example 19.4] and [24,25] for the special case g i (X, P n−1 )). Moreover, it is shown in [24] that when X is a subscheme of P n−1 , the (pushforward of the) Chern-Schwartz-McPherson class in the Chow ring A * (P n−1 ), namely c SM (X), and hence the topological Euler characteristic, namely χ(X), are completely determined by appropriate projective degrees.…”

“…. By [25,Lemma A.11], we have that (Θ, Λ) is a regular value of the restriction of π 2 to Γ if and only if for any (x, y, Θ, Λ) in the fiber π −1 2 (x, y, Θ, Λ) ∩ Φ Γ , we have that D(x, y, Θ, Λ) = 0. Hence, for (Θ, Λ) ∈ C (n−i)×r Λ × C i×n Θ \∆ pr X we have that (Θ, Λ) is a regular value of the restriction of π 2 to Γ.…”

“…. By [25,Lemma A.11], we have that (Θ, Λ) is a regular value of the restriction of π 2 to Γ if and only if for any (x, y, Θ, Λ) in the fiber…”

“…Homotopies utilizing this structure have also been proposed [28] and comparisons between homotopy continuation and modular Gröbner basis methods have been performed [6]. Moreover, aspects of this problem have also been considered in the context of computing Euler characteristics [25,29] and Segre classes [19,24]. In addition to formulating an upper bound on Alt's problem, the other novelty of our approach is to provide various probabilistic features of an effective algorithmic solution via modular Gröbner basis computations.…”